Itô’s stochastic calculus: Its surprising power for ... · H. Kunita / Stochastic Processes and their Applications 120 (2010) 622–652 623 based on martingales, through Doob–Meyer’s

Stochastic Processes and their Applications 120 (2010) 622–652www.elsevier.com/locate/spa

Ito’s stochastic calculus: Its surprising powerfor applications

Hiroshi Kunita

Kyushu University

Available online 1 February 2010

Abstract

We trace Ito’s early work in the 1940s, concerning stochastic integrals, stochastic differential equations(SDEs) and Ito’s formula. Then we study its developments in the 1960s, combining it with martingaletheory. Finally, we review a surprising application of Ito’s formula in mathematical finance in the 1970s.Throughout the paper, we treat Ito’s jump SDEs driven by Brownian motions and Poisson random measures,as well as the well-known continuous SDEs driven by Brownian motions.c© 2010 Elsevier B.V. All rights reserved.

MSC: 60-03; 60H05; 60H30; 91B28

Keywords: Ito’s formula; Stochastic differential equation; Jump–diffusion; Black–Scholes equation; Merton’s equation

0. Introduction

This paper is written for Kiyosi Ito of blessed memory. Ito’s famous work on stochasticintegrals and stochastic differential equations started in 1942, when mathematicians in Japanwere completely isolated from the world because of the war. During that year, he wrote acolloquium report in Japanese, where he presented basic ideas for the study of diffusion processesand jump–diffusion processes. Most of them were completed and published in English during1944–1951.

Ito’s work was observed with keen interest in the 1960s both by probabilists and appliedmathematicians. Ito’s stochastic integrals based on Brownian motion were extended to those

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H. Kunita / Stochastic Processes and their Applications 120 (2010) 622–652 623

based on martingales, through Doob–Meyer’s decomposition of positive submartingales. Thefamous Ito formula was extended to semimartingales.

Linking up with martingale theory, Ito’s stochastic calculus became a very useful toolfor applied problems. In stochastic control and filtering problems, Ito’s stochastic differentialequations are employed as models of dynamical systems disturbed by noise. Then conditionaldistributions of the filtering are obtained by solving stochastic partial differential equations.

Another important and thorough application of Ito’s stochastic calculus took place inmathematical finance. In 1973, Ito’s formula was applied in a striking way to the pricing ofoptions by Black–Scholes and Merton. The martingale representation theorem was used forthe hedging problem of derivatives and Girsanov’s theorem was applied for constructing a riskneutral measure. These are three pieces of tool kits or three jewels in mathematical finance.

In the first part of this paper, we will look back at Ito’s early work on stochastic calculus. Itincludes the Levy–Ito decomposition of a Levy process and stochastic differential equationsbased on Levy processes. In Section 2, we will consider a combination of Ito’s work withmartingale theory. Then we will explain in detail Ito’s stochastic integral based on a Poissonrandom measure. In Section 3, we will trace an application of Ito’s theory in mathematicalfinance. We will review the historical work by Black–Scholes and Merton on pricing optionsboth for diffusion and jump–diffusion models.

The author was a student of K. Ito at Kyoto in 1958–1961. Ito always loved and encouragedstudents and young mathematicians. He recommended us to study an important but difficultpaper. Following his suggestion, I read P. Levy’s paper on Markov chains (1951). Though Icould not understand the full importance of the paper, I got a small idea from the paper and wasable to finish the graduate course. I wish to express my cordial thanks to Kiyosi Ito.

1. Ito’s early work

1.1. The Levy–Ito decomposition of a Levy process

Ito’s first work [7] was published in 1942. The title is “On stochastic processes”. In those days,a stochastic process was often understood as a family of multivariate distributions. A Markovprocess was considered as a system of transition probabilities. After the fundamental work ofKolmogorov [21], 1933, the view of stochastic processes was changed greatly. Doob and Itodefined them as a family of random variables X t , t ∈ T on an infinite-dimensional probabilityspace, whose finite-dimensional distributions coincide with given multivariate distributions.

In the first half of the paper, Ito showed how we can construct a stochastic process called aLevy process associated with a given family of infinitely divisible distributions. In the secondhalf, he discussed the decomposition of a Levy process. Let X t , t ∈ T = [0, T ] (0 < T < ∞)be an R-valued stochastic process (where R is a Euclidean space) on the probability space(Ω ,F , P). It is called a Levy process if the following three properties are satisfied:

(1) X0 = 0 a.s. Sample paths are right continuous with left-hand limits a.s.(2) Let X t− = limε↓0 X t−ε . Then X t = X t− holds a.s. for any t .(3) It has independent increments, i.e., for any a ≤ t0 < t1 < · · · < tn ≤ b, the random variables

X ti − X ti−1 , i = 1, . . . , n are independent.

If the law of X t+h − Xs+h does not depend on h for any s, t , the Levy process is calledhomogeneous. (Recently, a homogeneous Levy process is simply called a Levy process). Ifthe sample paths X t are continuous a.s., the Levy process X t is called a Brownian motion.

624 H. Kunita / Stochastic Processes and their Applications 120 (2010) 622–652

In particular a homogeneous Brownian motion Wt with mean 0 and covariance t I is called astandard Brownian motion.

Ito looked into the sample paths of a Levy process profoundly and showed that any Levyprocess consists of two components: a Brownian motion and a Poisson random measure. Theidea goes back to P. Levy. His intuitive work became clear through Ito’s work.

Before the description of Ito’s work, we will define a Poisson random measure on T0 × R0,where T0 = T − 0 and R0 = R − 0. Let B be the Borel field of T0 × R0 and let n(dtdu) bea σ -finite measure on T0 × R0. A family of random variables N (B), B ∈ B is called a Poissonrandom measure on T0 × R0 with intensity measure n if

(1) For every B such that 0 < n(B) <∞, N (B) is Poisson distributed with mean n(B).(2) If B1, . . . , Bn are disjoint and 0 < n(Bi ) < ∞, i = 1, . . . , n, N (B1), . . . , N (Bn) are

independent.(3) For every ω, N (·, ω) is a counting measure on T0 × R0.

Now for a given Levy process X t , we define a random counting measure N on T0 × R0 by

N ((s, t] × A) = ]r ∈ (s, t];1Xr ∈ A, (1.1)

where 1Xr = Xr − Xr−. We set n(B) = E[N (B)] and define a compensated random measureby

N (dsdz) = N (dsdz)− n(dsdz). (1.2)

Theorem 1.1. Let X t be any Levy process.

(1) The counting measure N of (1.1) is a Poisson random measure on T0 × R0 with intensity

measure n satisfying∫ T

0

∫R0

|z|2

1+|z|2n(dsdz) <∞.

(2) For almost all ω,

X1t =

∫ t

0

∫|z|>1

zN (dsdz)+ limn→∞

∫ t

0

∫1n<|z|≤1

z N (dsdz), (1.3)

is defined for all t and the convergence is uniform with respect to t .(3) Wt := X t − X1

t is a Brownian motion independent of the Poisson random measure N (dtdz).

The last term of (1.3) is denoted by∫ t

0

∫0<|z|≤1 z N (dsdz).

Thus a Brownian motion and a Poisson random measure are the fundamental ingredientsof any Levy process, and furthermore the jumps of a Levy process X t are characterized by aPoisson counting measure N (dtdz). A surprising fact which impresses us is that the class ofLevy processes is large and rich, since the properties of Poisson random measures are quitedifferent according to their intensity measures. Still now it is an attractive subject for research(Sato [37]).

A consequence of Theorem 1.1 is that the law of a Levy process X t is infinitely divisible andits characteristic function E[ei(α,X t )] is given by the Levy–Khinchine formula.

exp

i(α, b(t))−12(α, A(t)α)+

∫ t

0

∫|z|>1

(ei(α,z)− 1)n(dsdz)

+

∫ t

0

∫0<|z|≤1

(ei(α,z)− 1− i(α, z))n(dsdz)

,


where b(t) and A(t) are the mean and the covariance of the Brownian motion Wt and n(dsdz) isthe intensity measure of the Poisson random measure N . These satisfy the following conditions:

(1) b(t) is continuous and b(0) = 0.(2) A(t) is continuous, increasing and A(0) = 0.

(3) n(dsdz) is a measure on T0 × R0 such that∫ T

0

∫R0

|z|2

1+|z|2n(dsdz) <∞ and n(t × R0) = 0

for any t .

The triplet (b(t), A(t), n(dtdz)) is the characteristic triplet of the Levy process. It determinesthe characteristic functions of X t so that it determines the laws of X t .

A Levy process X t is homogeneous if and only if its characteristic triplet is given by

(bt, At, dtν(dz)) where b, A are constants and ν is a measure on R0 satisfying∫|z|2

1+|z|2ν(dz) <

∞. The measure ν is called Levy measure.The proof of Theorem 1.1 is by no means simple. A detailed exposition may be found in Ito’s

lecture notes from Aarhus University [8]. Difficult parts of the proof would be: (a) N in (1.1) isa Poisson random measure, (b) It is independent of the Brownian motion Wt and (c) the uniformconvergence of (1.3) for almost all ω. Instead of (c) the following weaker assertion can be easilyverified. (c’) The convergence of (1.3) is uniform with respect to t in probability.

An alternative proof of (a) and (b) by a martingale method is given in [23].

1.2. Ito’s program for diffusions and jump–diffusions

After the work on Levy processes, Ito’s interest moved towards Markov processes. At thattime Kolmogorov’s work on diffusion processes [22] was known. He showed by an analyticalmethod that the transition probabilities of a diffusion process X t = (X1

t , . . . , Xdt ) on a

Euclidean space satisfy a linear second order parabolic partial differential equation which iscalled Kolmogorov’s equation. The equation tells us that the coefficients ai j (t, x) of the secondorder part and bi (t, x) of the first order part of the equation are characterized by

(a) E[X it+h − X i

t |X t = x] = bi (t, x)h + o(h), (1.4)

(b) E[(X it+h − X i

t )(Xjt+h − X j

t )|X t = x] = ai j (t, x)h + o(h).

In words, the transition probabilities of a diffusion process should be determined by the aboveinfinitesimal mean b(t, x) = (b1(t, x), . . . , bd(t, x)) and the infinitesimal covariance A(t, x) =(ai j (t, x)) of the diffusion process.

Again, Ito was interested in the sample paths of a diffusion process. His idea may be describedas follows. The above formula indicates that for each point (t, x), a Brownian motion with meanb(t, x) and covariance A(t, x) is tangent to the diffusion X t . The tangential Brownian motion atthe point (t, x) denoted by Z (t,x)(τ ), τ ≥ t could be written as

Z (t,x)(τ ) = b(t, x)(τ − t)+ σ(t, x)(Wτ −Wt ),

where σ(t, x) is the square root of A(t, x) and Wτ is a standard Brownian motion. Then thepaths of the diffusion process should be obtained by integrating the random tangential vectorfields Z(x, dt) := Z (t,x)(dt). Therefore the solution X t of the stochastic differential equation

dX t = Z(X t , dt) = b(t, X t )dt + σ(t, X t )dWt (1.5)

should be a diffusion process satisfying (1.4).


To make the idea rigorous, he defined stochastic integrals∫ t

0 fsdWs based on the Brownianmotion Wt and then solved Eq. (1.5). Ito’s stochastic integral is not similar to the Lebesgueintegral. He obtained a chain rule for stochastic integrals, which is now well known as Ito’sformula. Ito showed that the solution X t of Eq. (1.5) is a Markov process and its transitionprobabilities satisfy Kolmogorov’s forward equation.

Ito’s first work on this subject appeared during the second world war in 1942. It was acolloquium report at Osaka University, hand-written in Japanese. Later it was translated toEnglish [9]. In the report, he defined the stochastic integral based on a Brownian motion andgave some formulas concerning the calculus of stochastic integrals, which differs from usualcalculus. A formula given by him is:∫ t

0F ′(Ws)dWs = F(Wt )− F(W0)−

12

∫ t

0F ′′(Ws)ds,

where Wt is a Brownian motion and F is a C2-function. It was the first version of Ito’s formula.Then he considered the stochastic differential (1.5). Assuming the Lipschitz continuity for thecoefficients, he showed the existence and uniqueness of the solution and then the Markovproperty. Thus the solution is a diffusion process.

He was also interested in jump–diffusions. If X t is a jump–diffusion, the tangential processat (t, x) should be a homogeneous Levy process Z (t,x)(τ ), τ ≥ t with (b(t, x), A(t, x), dτν(t,x)

(dz)) as its characteristics. It is known that for the family of Levy measures ν(t,x)(dz), (t, x) ∈T × Rd , there exists a common Levy measure ν and a family of maps g(t,x) : R0 → R0,measurable with respect to (t, x, z), such that ν(t,x) = (g(t,x))∗ν, i.e., ν(t,x)(E) = ν(z :

g(t,x)(z) ∈ E) holds for any Borel set E . The maps g(t,x) satisfy∫|g(t,x)(z)|2

1+|g(t,x)(z)|2ν(dz) < ∞.

Common Levy measures are not unique, obviously. Ito took ν(dz) = 1|z|2

dz in the one-

dimensional case and Stroock [38] took ν(dz) = 1|z|d+1 dz in the d-dimensional case.

In any case we can define a family of tangential homogeneous Levy processes by a pair ofa standard Brownian motion and a homogeneous Poisson random measure with Levy measureν(dz). Indeed, let N (dτdz) be a homogeneous Poisson random measure with Levy measureν(dz), which is independent of a standard Brownian motion Wt . Set N (t,x)

:= (g(t,x))∗N . Thenit is a homogeneous Poisson random measure with the Levy measure ν(t,x)(dz). It holds∫ τ

t

∫|z′|≤1

z′ N (t,x)(dsdz′) =∫ τ

t

∫|g(t,x)|≤1

g(t,x)(z)N (dsdz).

Therefore the family of homogeneous Levy processes Z (t,x)(τ ), τ ≥ t with characteristics(b(t, x), A(t, x), dτν(t,x)(dz)) can be given by making use of a common Brownian motion Wτ

and a common Poisson random measure N (dτdz) as

Z (t,x)(τ ) = b(t, x)(τ − t)+ σ(t, x)(Wτ −Wt )

+

∫ τ

t

∫|g(t,x)|≤1

g(t,x)(z)N (dsdz)+∫ τ

t

∫|g(t,x)|>1

g(t,x)(z)N (dsdz).

Then the jump–diffusion should be obtained by integrating the above random vector fieldsZ(x, dt) := Z (t,x)(dt), i.e., the solution of the stochastic integral equation

dX t = b(t, X t−)dt + σ(t, X t−)dWt


+

∫R0

g1(t, X t−, z)N (dtdz)+∫

R0

g2(t, X t−, z)N (dtdz). (1.6)

Here g1(t, x, z) = g(t, x, z)1|g(t,x,z)|≤1 and g2(t, x, z) = g(t, x, z)1|g(t,x,z)|>1, whereg(t, x, z) = g(t,x)(z).

The colloquium report was not a final one. Some discussions were insufficient. However, mostof the work was completed and published later in English in a series of papers [10–14] and as amonograph [15].

1.3. Stochastic integrals based on Brownian motion and Ito’s formula

Stochastic integrals based on a Brownian motion are discussed in [10,13]. In the definition,Ito neither used terms such as filtration nor sub-σ -field nor martingale. Instead, he introducedthe property α. Let ξ = ξλ(t), λ ∈ Λ and η = ηµ(t);µ ∈ M be two families of stochasticprocesses. ξ is said to have the property α with respect to η, if for any t the following two familiesof random variables are independent:

ξλ(τ ), λ ∈ Λ, ηµ(τ ), µ ∈ M, 0 ≤ τ ≤ t, ηµ(σ )− ηµ(t);µ ∈ M, t ≤ σ .

Let Wt be a one-dimensional standard Brownian motion and ξ(t) be a measurable process, whichhas the property α with respect to Wt and satisfies

∫ t0 ξ(τ )

2dτ < ∞ a.s. for any t . Ito defined

the stochastic integral∫ t

0 ξ(τ )dWτ . If ξ(τ ) is in L2, i.e., E[∫ T

0 ξ(τ )2dτ ] < ∞, his definitionof the stochastic integral

∫ t0 ξ(τ )dWτ is exactly the same as that used nowadays. Indeed, if we

introduce the filtration Ft by Ft = σ(ξτ ,Wτ , 0 ≤ τ ≤ t), then Wt is an Ft -Brownian motionand ξ(t) is a measurable Ft -adapted process. Ito’s stochastic integral is then a square-integrableFt -martingale.

Further, if ξ(t) does not satisfy the above square integrability condition, Ito introduces asequence of processes ξn(t) by

ξn(t) = φn

(∫ t

0ξ(τ )2dτ

)ξ(t)

where φn(λ) = 1 if |λ| ≤ n and φn(λ) = 0 if |λ| > n. Then the stochastic integral for ξ wasdefined by∫ t

0ξ(τ )dWτ = lim

n→∞

∫ t

0ξn(τ )dWτ ,

when the limit exists in probability uniformly in t ∈ [0, T ]. Later in 1965, Ito and Watanabe [18]introduced the notion of a local martingale. The above Ito stochastic integral is indeed a localmartingale.

Now let Wt = (W 1t , . . . ,W r

t ) be an r -dimensional standard Brownian motion. Suppose thatthe family of measurable processes

β i (t), αij (t), i = 1, . . . , d, j = 1, . . . , r

has the property α with respect to the Brownian motion Wt . Suppose furthermore:∫ T

0 |βi (t)|dt <

∞,∫ T

0 |αij (t)|

2dt <∞.We may define a d-dimensional stochastic process X t = (X1t , . . . , Xd

t )


by

X it − X i

s =

∫ t

sβ i (τ )dτ +

∑j

∫ t

sαi

j (τ )dW jτ , 0 ≤ s ≤ t ≤ T, 1 ≤ i ≤ d.

It is simply written as

dX t = β(t)dt + α(t)dWt .

The stochastic process X t is now called an Ito process. In [13] he showed the celebrated Itoformula.

Theorem 1.2. Let F(t, x1, . . . , xd) be a function of class C1,2. Then η(t) = F(t, X t ) satisfies

dηt =

(Ft (t, X t )+

∑i

Fxi (t, X t )βi (t)

+12

∑i, j

Fxi x j (t, X t )(∑

k

αik(t)α

jk (t)

))dt +

∑i, j

Fxi (t, X t )αij (t)dW j

t ,

where Ft , Fxi are the partial derivatives of F with respect to t and xi , respectively and Fxi x j arethe second order partial derivatives of F with respect to xi , x j .

Ito’s formula is a change of variable formula or a chain rule for the calculus of stochasticintegrals. It has been applied to many types of stochastic calculus. Another important value ofIto’s formula is that we may find an explicit form of the generator of a diffusion process throughIto’s formula. It will be discussed in the next subsection.

1.4. Stochastic differential equations

Stochastic differential equations (SDE) (1.5) are studied in [11,12,15,13,14]. A continuousstochastic process X t with values in Rd is called a solution of Eq. (1.5) if X t has the propertyα with respect to Wt and satisfies Eq. (1.5). Assuming that the coefficients bi (t, x), σ i

j (t, x) areof linear growth and Lipschitz continuous with respect to x , Ito proved the existence and theuniqueness of the solution X t by the method of successive approximations.

Let X (s,x)t , t ≥ s be the solution starting from x at time s. In view of the linear growth propertyof the coefficients, it has the following moment property: For any p ≥ 2, there exists a positiveconstant C p such that [23]

E[|X (s,x)t |p] ≤ C p(1+ |x |)p, 0 ≤ ∀s, t ≤ T, ∀x ∈ Rd .

The solution is a Markov process. Set

Ps,t (x, E) := P(X (s,x)t ∈ E), Ps,t f (x) :=∫

Ps,t (x, dy) f (y).

Then Ps,t (x, E) are transition probabilities and satisfy the Chapman–Kolmogorov equation

Ps,t Pt,u f (x) = Ps,u f (x), ∀s < t < u

for any measurable function of polynomial growth.


Let f be a C2 function such that f, fxi , fxi x j are of polynomial growth. Set F = f and applyIto’s formula. Then we have

f (X t ) = f (x)+∫ t

sL(u) f (Xu)du +

∑i,k

∫ t

sfxi (u, Xu)σ

ik (u, Xu)dW k

u , (1.7)

where

L(t) f (x) =12

∑i, j

ai j (t, x) fxi x j (x)+∑

i

bi (t, x) fxi (x), (1.8)

and ai j (t, x) =∑

k σik (t, x)σ j

k (t, x). Each of the above terms in (1.7) is integrable. Theexpectation of the last term of (1.7) is 0. Then we find that Ps,t satisfies the equality

Ps,t f = f +∫ t

sPs,u L(u) f du. (1.9)

Differentiating each term of the above equality (1.9) with respect to t , we get Kolmogorov’sforward equation

∂

∂tPs,t f (x) = Ps,t L(t) f (x). (1.10)

The differential operator (1.8) is called the generator of the diffusion process.

Now, we will fix a point x ′ ∈ Rd and set X t+h = X (t,x′)

t+h . Eq. (1.9) implies

limh↓0

1h(E[ f (X t,t+h)] − f (x ′)) = L(t) f (x ′), (1.11)

for any C2 function f of polynomial growth. Take f (x) = xi − x ′i . Then L(t) f = bi (t).Eq. (1.11) is written as

limh↓0

1h

E[X it+h − x ′i ] = bi (t, x ′). (1.12)

Take f (x) = (xi − x ′i )(x j − x ′j ). Then L(t) f = ai j (t)+ (x j − x ′j )bi (t)+ (xi − x ′i )b

j (t). Then(1.11) implies

limh↓0

1h

E[(X it+h − x ′i )(X

jt+h − x ′j )] = ai j (t, x ′). (1.13)

The Eqs. (1.12) and (1.13) are equivalent to Eq. (1.4). Hence the solution of the SDE is exactlywhat Ito wanted to construct.

In [12] and [14], he studied stochastic differential equations on a differentiable manifold. Theequation is written in each local coordinate neighborhood using local coordinates. He explainedclearly how the coefficients bi and σ i

j of the equation should be changed according to the changeof the local coordinates. Then, assuming that the coefficients are uniformly bounded in anycanonical coordinates, he showed the existence and the uniqueness of the solution.

In [15], stochastic differential equations with jumps are discussed. The equations can bewritten as in (1.6). Ito studied the one-dimensional equation in the case where the Levy measureis given by 1

|z|2dz. He defined stochastic integrals based on the Poisson random measure N (dtdz)


and the compensated Poisson random measure N (dtdz), and then proved the existence, theuniqueness and the Markov property of the solution, under some regularity condition for thecoefficients b, σ, g1, g2. We will return to Eq. (1.6) in Section 2.

Ito’s aim of introducing stochastic differential equations was the construction of Markovdiffusion processes. After the 1960s, stochastic differential equations and diffusion processeswere studied in great detail. Topics include weak and strong solutions of SDE, SDE withboundary conditions, stochastic flow of diffeomorphisms, Malliavin calculus for SDE, etc. Asurvey of work on these topics is beyond the scope of this paper. We refer to Watanabe’scomprehensive paper [40] on these subjects. See also [19,36,39]. Further, a different approachusing Dirichlet forms was taken by Fukushima et al. [4] for the study of diffusions andjump–diffusions with the symmetry property.

Around 1960, stochastic differential equations were applied to engineering problems. Instochastic control and filtering problems, SDEs were taken as a model of a dynamical system,which is disturbed by noise. A dynamical system is supposed to move according to a differentialequation dX t = b(t,X)dt , where X = Xs; s ∈ T . But it is actually disturbed by a randomnoise σ(t,X)dWt . Its motion is then described by a SDE

dX t = b(t,X)dt + σ(t,X)dWt .

The coefficients b(t), σ (t) may depend on the whole past of the solution (Xs; s ≤ t). Thus thesolution is no longer a Markov process.

Later in the 1970s, the dynamic behavior of an asset price St was described by a SDE

dSt

St= b(t, St )dt + σ(t, St )dWt .

Then Ito’s stochastic calculus encountered with new big objects, which Ito did not design andwhich he even might not have wanted to. It seems that he was not very much interested inmathematical finance and he was not involved in it, either.

In Section 3, we will see how Ito’s formula is applied for pricing options.

2. Reinforcement of Ito’s work with martingales

2.1. Stochastic integrals with respect to semimartingales and Ito’s formula

In the 1960s, Ito’s theory on stochastic calculus was reinforced for martingales. Incombination with martingale theory, Ito’s stochastic calculus turned out to be a powerful toolnot only for mathematical problems related to stochastic analysis but also for applications toengineering and financial problems.

In 1962–1963, Meyer [30,31] showed that any positive submartingale can be decomposedinto the sum of a martingale and of a natural (predictable) increasing process, which is nowcalled the Doob–Meyer decomposition. The decomposition theorem was then applied to thestochastic integrals based on martingales, and Ito’s formula was extended to a chain rule forsemimartingales, by Kunita–Watanabe [26] and Meyer [32] in 1965–1967.

Let (Ω ,F , P) be a complete probability space. Let Ft , t ∈ [0, T ] be a family of sub-σ -fields of F . It is called a filtration if Fs ⊂ Ft holds for any s < t . A filtration is said to satisfy theusual conditions if F0 contains all null sets of F and satisfies Ft = ∧ε>0 Ft+ε (right continuous).In this subsection, we will consider problems on a probability space equipped with a filtrationFt with the usual conditions.


Filtrations play an important role in martingale theory and the theory of Markov processes.Further, in applications to engineering problems such as stochastic control and filtering or tomathematical finance, each Ft is understood as available data up to time t of a random timeevolution that we have been observing. Therefore the analysis of the filtration is important inapplications, too.

A stochastic process X t , t ∈ [0, T ] is said to be adapted (to Ft ) if X t is Ft measurable forany t . It is called progressively measurable if for every t , the map X : [0, t] × Ω 3 (u, ω) →X (u, ω) is B[0, t]×Ft -measurable, where B[0, t] is the Borel field of [0, t]. Let P be the smallestσ -field on [0,∞)×Ω with respect to which all left continuous adapted processes are measurable.A stochastic process X t is said to be predictable if it is measurable with respect to P .

A real valued adapted process X t is called a submartingale if it is integrable for any t andsatisfies E[X t |Fs] ≥ Xs for any s < t , where E[X |G] denotes the conditional expectation of theintegrable random variable X with respect to the sub-σ -field G of F . If the equality holds for anyt, s with t > s, X t is called a martingale. Let τ be a nonnegative random variable. It is called astopping time if τ ≤ t ∈ Ft holds for any t > 0.

Doob studied martingales in detail. The beautiful classical theory of martingales is found inthe books [3,5,35]. Doob showed that any martingale has a modification such that its samplepaths are right continuous with left-hand limits. In our discussion we will always consider sucha modification.

In [3], Doob pointed out that Ito’s stochastic integral can be extended to certain martingales.Let Mt be a square-integrable martingale. Suppose that there exists a right continuous(deterministic) increasing function F satisfying

E[(Mt − Ms)2|Fs] = F(t)− F(s), ∀s < t. (2.1)

Let φ(t) be a progressively measurable process with E[∫ T

0 |φ(t)|2dF(t)] < ∞. Doob showed

that the stochastic integral Nt =∫ t

0 φ(s)dMs is well defined for any t and it is a square-integrablemartingale.

If Mt is a square-integrable Levy process with mean 0, it is a martingale satisfying (2.1),where F(t) = at is the variance of Mt . We will give later an example of a square-integrablemartingale which is not a Levy process but satisfies (2.1) with a suitable F . However, if Mtis not a process with independent increments, we cannot expect in general that there exists afunction F satisfying (2.1).

Meyer [30,31] showed that if X t is a positive submartingale there exists a unique predictableincreasing process At such that X t − At is a martingale. Now let Mt be a square-integrablemartingale. Then M2

t is a submartingale. We denote the associated Ft -predictable increasingprocess At by 〈M〉t . Then the martingale Mt satisfies the equality

E[(Mt − Ms)2|Fs] = E[〈M〉t − 〈M〉s |Fs], ∀s < t. (2.2)

A similar equality had been known for additive functionals of a Markov process ξt . Here aprocess X t is called an additive functional if it satisfies the equality X t+s = Xs + X t θs for anys, t > 0 a.s., where θs is the shift of ξt such that ξt θs = ξs+t holds for any t . It tells us thatif Mt is a square-integrable martingale additive functional of a Markov process ξt , there existsa unique continuous increasing additive functional 〈M〉t satisfying (2.2). Using this equality,Motoo–Watanabe [33] defined the stochastic integral

∫ t0 φ(ξs)dMs as a continuous martingale

additive functional. The integral was then extended to arbitrary square-integrable martingalesin [26].


Following [26,32], we shall define the stochastic integral based on Mt for a predictable processφt whose norm

‖φ‖〈M〉 := E[∫ T

0|φ(t)|2d〈M〉t

]1/2

is finite. Let φ(t) =∑m

i=1 φi 1(ti−1,ti ](t) be a simple process such that each φi is bounded Fti−1 -measurable, where 0 = t0 < t1 < · · · < tm = T . Then the stochastic integral of φ with respectto dMt is defined by∫ t

0φdM :=

m∑i=1

φi (Mt∧ti − Mt∧ti−1).

It is a square-integrable martingale and satisfies E[|∫ T

0 φdM |2] = ‖φ‖2〈M〉. Next let φt be a

predictable process such that ‖φ‖〈M〉 <∞. Then there exists a sequence of simple processes φ(n)

such that ‖φ−φ(n)‖〈M〉→ 0. Then the sequence of stochastic integrals ∫ t

0 φ(n)dM is an L2(P)-

Cauchy sequence. The limit is a square-integrable martingale. We denote it by∫ t

0 φ(s)dMs or∫ t0 φdM .

Local martingales were defined by Ito–Watanabe [18]. Let X t be a right continuous adaptedprocess. It is called a local (or locally square-integrable) martingale if there exists an increasingsequence of stopping times τn such that τn → ∞ and the stopped processes Mt∧τn are (square-integrable) martingales. For a locally square-integrable martingale Mt , there exists a uniquepredictable increasing process 〈M〉t such that M2

t − 〈M〉t is a local martingale.Stochastic integrals are extended to a locally square-integrable martingale Mt and a pre-

dictable process φ(t) such that∫ T

0 |φ(s)|2d〈M〉s < ∞. In fact, there exists a locally square-

integrable martingale It (φ) such that the equality It∧τ (φ) =∫ t

0 φdMτ holds for anystopping time τ such that Mτ

t := Mt∧τ is a square-integrable martingale and satisfies E[∫ τ

0 (1+φ(s)2)d〈M〉s] <∞. We denote such It (φ) by

∫ t0 φdM .

In the following, a filtration Ft is fixed. The Ft -martingales are simply called martingales.A right continuous adapted process At is called a process of finite variation if it is written

as a difference of two adapted increasing processes. A right continuous adapted process X t iscalled a semimartingale if it is written as a sum of a locally square-integrable martingale Mt anda process of finite variation At . It is called a continuous semimartingale if both At and Mt arecontinuous processes. Let φ(t) be a predictable process such that

∫ t0 φ(s)dAs (Lebesgue integral)

and∫ t

0 φ(s)dMs are well defined a.s. The sum of these two integrals is denoted by∫ t

0 φ(s)dXs

or simply as∫ t

0 φdX . It is again a semimartingale.The bracket process for two square-integrable martingales was introduced in [33] for the study

of additive functionals of a Markov process. The authors defined the orthogonality of square-integrable martingales and discussed the orthogonal decomposition of a martingale. We willrecall their arguments. Let Mt and Nt be locally square-integrable martingales. Their bracketprocess is defined by 〈M, N 〉t = 1

4 〈M + N 〉t − 〈M − N 〉t . Then we have the relation⟨∫φdX,

∫ϕdY

⟩t=

∫ t

0φϕd〈X, Y 〉s . (2.3)

Two locally square-integrable martingales M, N are called orthogonal if 〈M, N 〉t = 0 holdsfor any t a.s. We denote by Mc

loc the set of all continuous locally square-integrable martingales


and Mdloc the set of all locally square-integrable martingales which are orthogonal to any

element of Mcloc. Elements of Md

loc are called purely discontinuous. For any locally square-integrable martingale Mt , there exist Mc

t ∈ Mcloc and Md

t ∈ Mdloc, and Mt is decomposed as

Mt = Mct + Md

t . Such a decomposition is unique.Let X t be a semimartingale. Then its sample paths are right continuous with left-hand limits.

We denote the left limits of X t by X t−. Then X t− is a left continuous predictable process.For a semimartingale X t , we may define the stochastic integral

∫ t0 Xs−dXs . Another bracket

process [X ]t was introduced in [32]. It is defined by [X ]t = X2t − 2

∫ t0 Xs−dXs . We define

[X, Y ] = 14 [X + Y ] − [X − Y ].

Let X, Y be semimartingales. They are decomposed as X t = Mct + Md

t + At and Yt =

N ct + N d

t + Bt , where Mct , N c

t ∈ Mcloc, Md

t , N dt ∈ Md

loc and At , Bt are processes of finitevariation. We have the formula

[X, Y ]t = 〈Mc, N c〉t +

∑s≤t(1Xs)(1Ys). (2.4)

Hence [X, Y ]t is a process of finite variation. Its continuous part [X, Y ]ct coincides with〈Mc, N c

〉t .Ito’s formula can be extended to semimartingales. A d-dimensional stochastic process

X t = (X1t , . . . , Xd

t ) is called a semimartingale if all its components X it , i = 1, . . . , d are

semimartingales. When a semimartingale has jumps, several versions of the Ito formula are used.The following is due to Meyer [32,34].

Theorem 2.1. Let X t = (X1t , . . . , Xd

t ) be a semimartingale. Let F(t, x) be a function of theC1,2-class. Then F(t, X t ) is again a semimartingale and the following formula holds:

F(t, X t ) = F(0, X0)+

∫ t

0Ft (s, Xs−)ds +

∑i

∫ t

0Fxi (s, Xs−)dX i

s

+12

∑i, j

∫ t

0Fxi x j (s, Xs−)d[X i , X j

]cs

+

∑0<s≤t

F(s, Xs)− F(s, Xs−)−

∑i

Fxi (s, Xs−)1X is

. (2.5)

2.2. Stochastic integrals based on compensated Poisson random measure

Let (Ω ,F , P) be a probability space, where a standard m-dimensional Brownian motionWt = (W 1

t , . . . ,W mt ) and a homogeneous Poisson random measure N (dtdz) on T0 × R0 are

defined, T0 = [0, T ] − 0 and R0 = Rd− 0. These two processes are assumed to be

independent. Let B be the Borel field of R0. For B ∈ B, set Nt (B) = N ((0, t] × B). If it isintegrable for any t , it is a Poisson process. Its mean is written as tν(B) for a measure ν, calledthe Levy measure. Let Ft be the smallest filtration such that for any t , the family of randomvariables Ws, Ns(B), B ∈ B, s ≤ t are measurable with respect to Ft . In Sections 2.2–2.4, wewill consider problems under this filtration.

Let N (dsdz) = N (dsdz) − dsν(dz) be the compensated Poisson random measure. Supposeν(B) < ∞. Then Nt (B) = N ((0, t] × B) = Nt (B) − tν(B) is a compensated Poisson


process. Its mean is 0 and its covariance tν(B). Hence it is a square-integrable martingale with〈N (B)〉t = tν(B).

We shall define a stochastic integral based on the compensated Poisson random measure. Arandom function ψ(t, z, ω), (t, z, ω) ∈ T0× R0×Ω is called a predictable process if it is P×B-measurable. We shall define a stochastic integral of the form

∫ t0

∫R0ψ(s, z)N (dsdz) or simply∫ t

0

∫ψdN for such ψ . We first introduce some functional spaces:

Ψ2loc =

ψ(t, z) predictable |

∫ T

0

∫R0

|ψ(t, z)|2dtν(dz) <∞,

Ψ2=

ψ(t, z) ∈ Ψ2

loc | E[∫ T

0

∫R0

|ψ(t, z)|2dtν(dz)]<∞

,

Ψloc =

ψ(t, z) predictable |

∫ T

0

∫R0

|ψ(t, z)|2

1+ |ψ(t, z)|dtν(dz) <∞

,

Ψ =ψ(t, z) ∈ Ψloc | E

[∫ T

0

∫R0

|ψ(t, z)|2

1+ |ψ(t, z)|dtν(dz)

]<∞

.

We denote by Ψ1loc the set of all ψ’s such that |ψ |1/2 ∈ Ψ2

loc. Ψ1 is defined similarly. Then allthe above spaces are vector spaces.

It holds Ψ1∪Ψ2

⊂ Ψ and Ψ1loc∪Ψ2

loc ⊂ Ψloc. Further, a predictable process ψ belongs to Ψ(or Ψloc) if and only if ψ1 := ψ1|ψ |>1 ∈ Ψ1 (or ∈ Ψ1

loc) and ψ2 := ψ1|ψ |≤1 ∈ Ψ2 (or ∈ Ψ2loc).

If ν is a finite measure, we have Ψ2⊂ Ψ1

= Ψ and Ψ2loc ⊂ Ψ1

loc = Ψloc.We set∫

R0

ψ(z)N ((s, t], dz) :=∫

R0

ψ(z)Nt (dz)−∫

R0

ψ(z)Ns(dz).

Lemma 2.2. Let ψ(z) be a Fs×B-measurable random variable such that E[∫

R0ψ(z)2ν(dz)] <

∞. Then it holds for any s < t ,

E[∫

R0

ψ(z)N ((s, t], dz)

∣∣∣∣Fs

]= 0, a.s. (2.6)

E[(∫

R0

ψ(z)N ((s, t], dz))2∣∣∣∣Fs

]= (t − s)

∫R0

ψ(z)2ν(dz), a.s. (2.7)

Proof. Since N ((s, t], dz) is independent of Fs , we have

E[∫

ψ(z)N ((s, t], dz)

∣∣∣∣Fs

]=

∫R0

ψ(z)E[N ((s, t], dz)] = 0, a.s.

Suppose thatψ(z) is a simple function written as∑n

i=1 ci 1Ai , where Ai , i = 1, . . . , n are disjointBorel sets in R0. Then

E[(∫

R0

ψ(z)N ((s, t], dz))2∣∣∣∣Fs

]=

∑i, j

ci c j E[N ((s, t] × Ai )N ((s, t] × A j )|Fs].

Note that N ((s, t] × Ai ), i = 1, . . . , n are Poisson random variables with respective intensities(t − s)ν(Ai ), and they are mutually independent and are also independent of Fs . Then the above


is equal to∑

i c2i (t− s)ν(Ai ) = (t− s)

∫ψ2ν(dz), showing (2.7). The equality (2.7) is extended

to arbitrary square integrable ψ .

Now letψ(t, z) =∑ψi (z)1(ti−1,ti )(t) be a simple predictable process, whereψi are Fti−1×B-

measurable functions with E[∫|ψi (z)|2ν(dz)] < ∞. The stochastic integral of ψ with respect

to dN is defined by∫ t

0

∫ψdN :=

∑i

∫ψi (z)N ((ti−1 ∧ t, ti ∧ t], dz).

It is a square-integrable martingale and the equality⟨∫ ∫ψdN

⟩t=

∫ t

0

∫R0

ψ(s, z)2dsν(dz) (2.8)

holds in view of the above lemma.For ψ(t, z) ∈ Ψ2, there exists a sequence ψn(t, x) of simple predictable processes such

that E[∫ T

0

∫R0|ψ − ψn|

2dtν(dz)]→ 0. Then the sequence of stochastic integrals

∫ t0

∫ψndN

converges to a square-integrable martingale Mt . We denote it by∫ t

0

∫ψdN .

The stochastic integral can be extended forψ ∈ Ψ2loc as a locally square-integrable martingale,

which we denote by∫ t

0

∫ψdN . Next let ψ ∈ Ψloc. Set ψ1 = ψ1|ψ |>1 and ψ2 = ψ1|ψ |≤1. Then

ψ2 ∈ Ψ2loc and the integral

∫ t0

∫ψ2dN is well defined as a locally square-integrable martingale.

For ψ1, both integrals∫ t

0

∫ψ1dN and

∫ t0

∫ψ1dsdν are well defined. Denote the difference of

the above two integrals by∫ t

0

∫ψ1dN . It is a local martingale. We define the stochastic integral∫ t

0

∫ψdN as the sum of the two integrals

∫ t0

∫ψi dN , i = 1, 2. It is a local martingale.

If ψ ∈ Ψ , then ψ2 ∈ Ψ2. The stochastic integral of ψ2 is a square-integrable martingale.Further since ψ1 ∈ Ψ1,

∫ t0

∫ψ1dN is a martingale. Hence the stochastic integral

∫ t0

∫ψdN is a

martingale.It is convenient to introduce the Levy process

Jt =

∫ t

0

∫0<|z|≤1

z N (dsdz)+∫ t

0

∫|z|>1

zN (dsdz).

Then we may rewrite the Poisson random measure as the counting measure of the jumps of Jt .We have∫ t

0

∫ψdN =

∑s≤t

ψ(s,1Js). (2.9)

We summarize the above theory on stochastic integrals based on compensated Poisson randommeasure.

Proposition 2.3. Suppose ψ ∈ Ψloc (or Ψ ). Then the stochastic integral∫ t

0

∫ψdN is defined as

a local martingale (or martingale). It holds∫ t

0

∫ψdN = lim

ε↓0

∑s≤t,|1Js |>ε

ψ(s,1Js)−

∫ t

0

∫|z|>ε

ψ(s, z)dsν(dz).

If ψ ∈ Ψ2loc (or Ψ2),

∫ t0

∫ψdN is a locally square-integrable (or square-integrable) martingale.


Now, for vector processes φ(t) = (φ1(t), . . . , φm(t)), we introduce

Φ2loc =

φ(t) predictable |

∫ T

0|φ(t)|2dt <∞

,

Φ2=

φ(t) ∈ Φ2

loc | E

[∫ T

0|φ(t)|2dt

]<∞

.

The stochastic integrals∫ t

0 φi dW i

s are well defined. These are locally square martingales ifφ ∈ Φ2

loc and are square-integrable martingales if φ ∈ Φ2.

Proposition 2.4. For φ, φ′ ∈ Φ2loc and ψ,ψ ′ ∈ Ψ2

loc, we set

Mt =∑

j

∫ t

0φ j dW j

s +

∫ t

0

∫ψdN , M ′t =

∑j

∫ t

0φ′ j dW j

s +

∫ t

0

∫ψ ′dN .

These are locally square-integrable martingales and

〈M,M ′〉t =∑

j

∫ t

0φ jφ

′ j ds +∫ t

0

∫ψψ ′dsdν, (2.10)

[M,M ′

]t =

∑j

∫ t

0φ jφ

′ j ds +∫ t

0

∫ψψ ′dN . (2.11)

2.3. Ito’s formula, martingale representation theorem and Girsanov’s theorem

We give three theorems, which are useful in applications.Let X t = (X1

t , . . . , Xdt ) be a semimartingale such that

X it − X i

s =

∫ t

sβ i (τ )dτ +

∑j

∫ t

sαi

j (τ )dW jτ +

∫ t

s

∫R0

γ i (τ, z)N (dτdz),

where β(t) ∈ Φ2loc and γ (t, z) ∈ Ψloc. We call it an Ito jump process. Ito’s formula (Theorem 2.1)

may be rewritten for Ito jump processes. The following is another version of Ito’s formula dueto [26,5]. The formula is useful for the study of jump–diffusions.

Theorem 2.5. Let X t = (X1t , . . . , Xd

t ) be an Ito jump process represented above. LetF(t, x1, . . . , xd) be a C1,2-function such that F

1+|x | is bounded. Then ηt = F(t, X t ) is an Itojump process and is decomposed as

dηt =

[Ft (t, X t−)+

∑i

Fxi (t, X t−)βi (t)+

12

∑i, j

Fxi x j (t, X t−)

(∑k

αik(t)α

jk (t)

)

+

∫R0

F(t, X t− + γ (t, z))− F(t, X t−)−

∑i

γ i (t, z)Fxi (X t−)

ν(dz)

]dt

+

∑i, j

Fxi (t, X t−)αij (t)dW j

t +

∫R0

F(t, X t− + γ (t, z))− F(t, X t−)N (dtdz).


Proof. In Theorem 2.1, we may rewrite

Fxi (t, X t−)dX it = Fxi (t, X t−)β

i (t)dt +∑

j

Fxi (t, X t−)αij (t)dW j

t

+

∫Fxi (t, X t−)γ

i (t, z)N (dtdz),

Fxi x j (t, X t−)d[X i , X j]ct = Fxi x j (t, X t−)

(∑k α

ik(t)α

jk (t)

)dt, because

[X i , X j]ct =

∑k,l

⟨∫αi

kdW k,

∫α

jl dW l

⟩t

=

∑k,l

∫ t

0αi

kαjl d〈W k,W l

〉s =∑

k

∫ t

0αi

kαjk ds.

Next, checking that

F(t, X t− + γ (t, z))− F(t, X t−) ∈ Ψ2loc,

∑i

γ i (t, z)Fxi (X t−) ∈ Ψ2loc,

G(t, z) := F(t, X t− + γ (t, z)) − F(t, X t−) −∑

i γi (t, z)Fxi (X t−) ∈ Ψ1

loc, and noting that1X t = γ (t,1Jt ), we have by Proposition 2.3,∑

0<s≤t

F(s, Xs)− F(s, Xs−)−

∑i

Fxi (s, Xs−)1X is

=

∫ t

0

∫G(s, z)N (dsdz)+

∫ t

0

(∫G(s, z)ν(dz)

)ds

=

∫ t

0

∫F(s, Xs− + γ (s, z))− F(s, Xs−)N (dsdz)

−

∑i

∫ t

0

∫Fxi (s, Xs−)γ

i (s, z)N (dsdz)+∫ t

0

(∫G(s, z)ν(dz)

)ds.

Therefore we get the formula.

Associated with the filtration Ft generated by the Brownian motion Wt and Poisson randommeasure N (dtdz), we have the martingale representation theorem

Theorem 2.6. ([26,24]) Let Mt be a locally square-integrable martingale. Then there existsφ(t) ∈ Φ2

loc and ψ(t, z) ∈ Ψ2loc and Mt is represented by

Mt − M0 =∑

i

∫ t

0φi dW i

+

∫ t

0

∫ψdN . (2.12)

Martingale representations are also known for processes which are not Levy processes.Here we shall consider one such martingale. Let Wt be a one-dimensional standard Brownianmotion and let γt = sups ≤ t;Ws = 0. We define µt := sgn(Wt )

√t − γt . Then µt is

a purely discontinuous martingale, called the Azema martingale. It holds 〈µ〉t = t/2. Hencethe martingale µt has the property (2.1) with F(t) = t/2. For the proof of the theorem, Ito’swork [16,17] is used. Let St be the filtration generated by µt . Then for any square-integrable


St -martingale Mt , there exists a predictable process φ with E[∫ T

0 φ(t)2dt] < ∞ and Mt isrepresented as

Mt = M0 +

∫ t

0φdµ.

For details we refer to Mansuy–Yor [28].Now, let αt be a local martingale such that αt > 0 a.s. for any t . It is called a positive local

martingale. We assume α0 = 1.

(1) ([24]) We may represent it as the solution of a linear SDE

dαt = αt−dZ t , (2.13)

where Z t is a local martingale represented as

Z t =∑

i

∫ t

0φi dW i

+

∫ t

0

∫ψdN , (2.14)

with φ ∈ Φ2loc and ψ ∈ Ψloc such that 1+ ψ(t, z) > 0 for any t, z.

(2) Conversely suppose we are given a pair of φ ∈ Φ2loc and ψ ∈ Ψloc such that ψ(t, z)+ 1 > 0

for any t, z. Define Z t by (2.14) and let αt = αt (φ, ψ) be the solution of (2.13). Then αt is apositive local martingale.

(3) If φ(t) and∫|ψ(t, z)|2ν(dz) are bounded processes, then αt is a martingale. This fact is

shown as follows. Set τn = inf0 < t < T ;αt ≥ n (=∞ if · · · = ∅). Then τn, n = 1, 2, . . .is an increasing sequence of stopping times such that τn → ∞ a.s. The stopped processesα(n)t = αt∧τn satisfy α(n)t − 1 =

∫ t∧τn0 α

(n)s−dZs . The stochastic integral in the right-hand side

is a square-integrable martingale, because its bracket process satisfies

E[⟨∫

α(n)s−dZs

⟩t∧τn

]= E

[∫ t∧τn

0(α(n)s− )

2d〈Z〉s]

= E[∫ t∧τn

0(α(n)s− )

2(φ(s)2 +

∫ψ(s, z)2ν(dz)

)ds]

≤ nK t <∞,

where K = sups,ω(|φ(s)|2+∫ψ(s, z)2ν(dz)). Therefore α

(n)t is a square-integrable

martingale. Further, the above computation yields another inequality

E[(α(n)t − 1)2] = E[⟨∫

α(n)s−dZs

⟩t∧τn

]≤ K

∫ t

0E[(α(n)s )2]ds.

Therefore supn supt≤T E[(α(n)t )2] < ∞. Then for any 0 ≤ t ≤ T , the sequence of random

variables α(n)t is uniformly integrable and hence it converges to αt in L1. Therefore αt is amartingale.

Now assume αt = αt (φ, ψ) is a martingale. Define

Q(B) = E[αT 1B], ∀B ∈ FT . (2.15)

Then Q is a probability measure on (Ω ,FT ). We denote it by αT · P .Since Q is equivalent (mutually absolutely continuous) to P , a stochastic process on

(Ω ,FT , P) can be regarded as a stochastic process on (Ω ,FT , Q). The following is a Girsanovtheorem for Ito jump processes.


Theorem 2.7. ([24]) With respect to Q, we have

(1) Wφt := Wt −

∫ t0 φ(s)ds is a standard Brownian motion.

(2) The compensator of N is nψ (dsdz) = (1+ ψ(s, z))dsν(dz), i.e.,

Nψt (g) :=

∫ t

0

∫R0

g(s, z)N (dsdz)− nψ (dsdz)

is a local martingale, if g(s, z) is square integrable with respect to nψ (dsdz) a.s.

2.4. SDE with jumps and jump–diffusions

In Section 1 we mentioned that Ito constructed a jump–diffusion by solving an SDE. His ideawas to integrate tangential Levy processes Z(x, dt), which consist of a Brownian motion Wt andof a Poisson random measure N (dtdz). In this subsection, we shall study his approach in detail.

Let us consider Eq. (1.6). Assume that b, σ are of linear growth and Lipschitz continuous.Assume further that g is of linear growth and Lipschitz continuous with respect to ν,

|g(t, x, z)| ≤ K (z)(1+ |x |), (2.16)

|g(t, x, z)− g(t, y, z)| ≤ L(z)|x − y|,

where∫(K (z)2 + L(z)2)ν(dz) <∞. Then we may rewrite Eq. (1.6) simply as

dX t = b(t, X t−)dt + σ(t, X t−)dWt +

∫R0

g(t, X t−, z)N (dtdz),

where the drift vector b(t, x) of the above equation is changed from the one in Eq. (1.6) byadding the term

∫|g(t,x,z)|>1 g(t, x, z)ν(dz).

More precisely, an Ft -semimartingale X t is called a solution of the equation starting fromX t0 at time t0 if it satisfies the equality

X t = X t0 +

∫ t

t0b(s, Xs−)ds +

∫ t

t0σ(s, Xs−)dWs

+

∫ t

t0

∫R0

g(s, Xs−, z)N (dsdz). (2.17)

The existence and the (pathwise) uniqueness of the solution may be shown under the aboveconditions for coefficients.

Let X (s,x)t be the solution starting from x at time s. We have the relation X (s,x)u = X(t,X (s,x)t )u

a.s. for s < t < u. Since X (t,x)u is independent of Ft and since X (s,x)t is Ft -measurable, we get

E[ f (X (s,x)u )|Ft ] = E[ f (X (t,y)u )]y=X (s,x)t

, a.s.

Therefore for any s, x , X (s,x)t is a Markov process. Set Ps,t (x, E) = P(X (s,x)t ∈ E) and denotePs,t f (x) =

∫f (y)Ps,t (x, dy). Then the above equality is written as

E[ f (X (s,x)u )|Ft ] = Pt,u f (X (s,x)t ), a.s.

Taking expectations, we get the equality Ps,u f (x) = Ps,t Pt,u f (x), which is called theChapman–Kolmogorov equation. The solution X (s,x)t is called a jump–diffusion.


We may obtain Kolmogorov’s forward and backward equations for its transition probabilitiesby applying Ito’s formula to jump–diffusion processes. We shall apply Ito’s formula(Theorem 2.5) to a C2-function f such that fxi , fxi x j are bounded. Set

L(t) f (x) : =12

∑i, j


i

bi (t, x) fxi (x)

+

∫R0

f (x + g(t, x, z))− f (x)−

∑i

gi (t, x, z) fxi (x)ν(dz). (2.18)

The last integral is well defined since

| f (x + g(t, x, z))− f (x)−∑

i

gi (t, x, z) fxi (x)| ≤12

∑i, j

‖ fxi x j ‖|g(t, x, z)|2

and g satisfies (2.16). Further, each term on the right-hand side of (2.18) is dominated byC(1+ |x |)2.

Now, define a family of measures by ν(t,x) = (g(t,x))∗ν, where g(t,x)(z) = g(t, x, z). Then(b(t, x), A(t, x), dτν(t,x)(dz)) is the characteristic triplet of the tangential Levy process at thepoint (t, x) for Eq. (2.17). The above operator L(t) can be rewritten by

L(t) f (x) =12

∑i, j


i

bi (t, x) fxi (x)

+

∫R0

f (x + y)− f (x)−

∑i

yi fxi (x)ν(t,x)(dy). (2.19)

For X t = X (s,x)t , Ito’s formula is written as

f (X t ) = f (x)+∫ t

sL(u) f (Xu−)du +

∑i, j

∫ t

sfxi (Xu−)σ

i j (u, Xu−)dW ju

+

∫ t

s

∫R0

f (Xu− + g(u, Xu−, z))− f (Xu−)N (dudz). (2.20)

We shall consider the expectation of each term of the above equation. We have E[ f (X t )] =

Ps,t f (x) and

E[∫ t

sL(u) f (Xu−)du

]=

∫ t

sE[L(u) f (Xu−)]du =

∫ t

sPs,u L(u) f (x)du.

Here we used that P(Xu = Xu−) = 1 holds for any u. Processes φ j (u) =∑

i fxi (Xu−)σi j

(u, Xu−), j = 1, . . . , d belong to Φ2. Then their stochastic integrals with respect to dW jt are

martingales with mean 0. Further, the process ψ(u, z) = f (Xu− + g(u, Xu−, z)) − f (Xu−)

satisfies

|ψ(u, z)|2 ≤(∑‖ fxi ‖

)2(1+ |Xu−|

2)K (z)2.


Hence ψ ∈ Ψ2 and∫ t

0

∫ψdN is a square-integrable martingale with mean 0. Then Eq. (2.20)

implies that the transition probabilities Ps,t satisfy

Ps,t f (x) = f (x)+∫ t

sPs,u L(u) f (x)du.

Consequently, we have

Theorem 2.8. Let X t be the jump–diffusion determined by SDE (2.17). Then its transitionprobabilities Ps,t satisfy Kolmogorov’s forward integro-differential equation

∂

∂tPs,t f (x) = Ps,t L(t) f (x), ∀0 < s < t < T . (2.21)

where L(t) f is given by (2.18) or (2.19).

The operator L(t) is called the generator of the jump–diffusion X t . To obtain Kolmogorov’sbackward equation, we need some smoothness conditions on the coefficients.

Theorem 2.9. Assume that the coefficients b, σ, g are twice continuously differentiable withrespect to x and their derivatives (up to the second order) are of linear growth and Lipschitzcontinuous. Then if f is a C2-function with bounded derivatives, Ps,t f (x) is a C1,2 function of(s, x). It satisfies Kolmogorov’s backward integro-differential equation

∂

∂sPs,t f (x)+ L(s)Ps,t f (x) = 0, ∀0 < s < t < T . (2.22)

Proof. 1 It is known that the solution X (s,x)t is twice continuously differentiable with respect tox (see [23]). Then Ps,t f (x) is also twice continuously differentiable with respect to x , if f is asmooth function. It is actually a C1,2 function of (s, x).

We will prove (2.22). For a fixed t , set Ms = Ps,t f (Xs), t0 ≤ s ≤ t , where Xs, t0 ≤ s ≤ t isa solution of the SDE. It is a martingale, because for t0 ≤ s ≤ u ≤ t ,

E[Mu |Fs] = E[Pu,t f (Xu)|Fs] = Ps,u Pu,t f (Xs) = Ps,t f (Xs) = Ms .

Applying Ito’s formula to the function F(s, x) := Ps,t f (x), we have

Mu = F(t0, X t0)+

∫ u

t0

∂∂s

F(s, Xs−)+ L(s)F(s, Xs−)

ds + a local martingale.

Therefore the integral∫ u

t0· · ·ds is a local martingale, so that it is equal to 0 a.s. Therefore

∂∂s + L(s)F(s, Xs−) = 0 a.s. for any t0 ≤ s ≤ t . Since this is valid for any solution Xs , we

have ∂∂s + L(s)F(s, x) = 0 for any t0 ≤ s ≤ t and x ∈ Rd .

Let us transform the probability measure P to an equivalent measure Q = αT (φ, ψ) · P .Assume that the pair (φ(t), ψ(t, z)) is given by φ(t) = φ(t, X t ) and ψ(t, z) = ψ(t, X t−, z),where φ(t, x), ψ(t, x, z) are continuous functions such that both φ(t, x) and

∫ψ2(t, x, z)ν(dz)

are bounded with respect to t, x . Then αt (φ, ψ) is a martingale. Hence Q = αT (φ, ψ) · P is welldefined.

1 An alternative and rigorous proof is given in [25], where the backward chain rule (backward Ito formula) for the

stochastic flow X (s,x)t is used.


We are interested in the change of generator for X through Girsanov’s transformation. We set

b(φ,ψ)(t, x) := b(t, x)+ φ(t, x)σ (t, x)+∫

R0

ψ(t, x, z)g(t, x, z)dν(z) (2.23)

and define

L Q(t) f (x) :=12

∑i, j


i

b(φ,ψ),i (t, x) fxi (x)

+

∫R0

(f (x + g(t, x, z))− f (x)−

∑i

gi (t, x, z) fxi (x))(1+ ψ(t, x, z))ν(dz). (2.24)

Each term of the above is well defined and is dominated by C(1+ |x |)2.

Theorem 2.10. X t is a jump–diffusion with respect to Q. Its generator is given by (2.24).

Proof. Let X t = X (s,x)t be the solution starting from x at time s. We can rewrite Ito’s formula(Theorem 2.5) as

f (X t ) = f (Xs)+

∫ t

sL Q(u) f (Xu−)du +

∑i, j

∫ t

sfxi (Xu−)σ

i j (u, Xu−)dWφ, ju

+

∫ t

s

∫ f (Xu− + g(u, Xu−, z))− f (Xu−)N

ψ (dudz).

Take the expectation of each term with respect to Q. The expectations of the last two stochasticintegrals with respect to dWφ and dNψ are both 0. Therefore we get the equality

P Qs,t f (x) = f (x)+

∫ t

sP Q

s,u L Q(u) f (x)du,

where P Qs,t f (x) = E Q

[ f (X (s,x)t )] =∫

f (X (s,x)t )dQ. Then we see that L Q(t) is the generator ofX t under Q.

3. Application to mathematical finance

3.1. Asset, option and non-arbitrage market

We will introduce two processes. One is the price of the riskless asset. It is defined byB0

t = er t , where r is a positive constant. It is called a bank account process. The other oneis the price process of a risky asset which may have jumps.

We need some notation. Let (Ω ,F , P) be a probability space, where the following twoprocesses are defined. Wt is a one-dimensional standard Brownian motion and N (dtdz) isa homogeneous Poisson random measure on R0 with intensity measure dtν(dz), which aremutually independent. Let Ft be the smallest filtration such that for any t , Ws, N (s, B), s ≤t, B ∈ B are measurable with respect to Ft . The price process St of a risky asset is defined bythe following one-dimensional SDE:

dSt = St−Y (St−, dt), (3.1)


where Y (x, dt) is a random vector field given by

Y (x, dt) = b(t, x)dt + σ(t, x)dWt +

∫R0

g(t, x, z)N (dtdz). (3.2)

For the coefficients b, σ, g, we assume the following:

(1) b, σ are bounded and Lipschitz continuous. Further, σ is uniformly positive.(2) g(t, x, z) is bounded and Lipschitz continuous with respect to ν. Further, it holds g(t, x, z)+

1 > 0 for any t, x, z.

Then Eq. (3.1) has a unique solution. Let Ss,t (x) be the solution starting from x > 0 at time s.Then it holds Ss,t (x) > 0 for any t > s a.s.

The random vector field Y (x, dt) is called the return process. Its mean b(t, x)dt shows theexpected rate of return of the asset and the variance (σ (t, x)2 +

∫g(t, x, z)2ν(dz))dt shows the

rate of risk of the asset. σ(t, x) is called the volatility.∫

g(t, x, z)2ν(dz) is the rate of jump risk.If the coefficients b, σ and g do not depend on the state x and time t , the random vector field

Y (x, dt) does not depend on x . It is a Levy process. Then log Ss,t (x) is also a Levy process. Thesolution Ss,t (x) is called a geometric Levy process. In particular if the Poisson part is 0 (g = 0),the solution is called a geometric Brownian motion or Black–Scholes model. If the Levy measureν has a finite total mass, it is called a Merton model.

A nonnegative FT -measurable random variable h is called a contingent claim. If it is writtenas h = h(ST ) with a fixed time T , it is called a European option and the function h is called apay-off function. Let K be a positive constant. In the case where h(x) = (x − K )+, it is called aEuropean call option with exercise price K and in the case where h(x) = (K − x)+, it is calleda European put option with exercise price K . In this paper, we assume that h is a continuousfunction and h(x)

1+|x | is bounded.We shall consider a European option. Suppose that we have a guarantee to be paid the amount

h(ST ) at time t = T . An important question in finance is to find the price Pt of the option ateach time 0 ≤ t < T , which both the buyer and the seller of the option can agree with. Theproblem was solved in 1973 by Black–Scholes and Merton in the case where the price process isa diffusion (Black–Scholes model). In 1975 Merton proposed a price in the case where the priceprocess is a jump–diffusion (Merton model). We will review their arguments from a slightlydifferent viewpoint.

We will assume that the price of the option is given in the form Pt = F(t, St ) with a functionF(t, x), t ∈ [0, T ], x > 0 of the C1,2-class such that F(T, x) = h(x). The function F is calleda pricing function. In their arguments, they applied Ito’s formula in a striking way. They found apartial differential equation or an integro-differential equation for the pricing function F(t, x) insuch a way that the market (B0

t , St , Pt ) becomes non-arbitrage, through Ito’s formula.Now we shall consider a market consisting of the triple (B0

t , St , Pt ). By a strategy or aportfolio, we mean a triple of progressively measurable processes θ(t) = (θ0(t), θ1(t), θ2(t)).The value process of the portfolio θ(t) is defined by

Vt (θ) = θ0(t)B0t + θ1(t)St + θ2(t)Pt . (3.3)

Suppose that∫ T

0 (|θ0| + |θ1|2+ |θ2|

2)dt <∞ and that the pair (θ1(t), θ2(t)) is predictable. If theequality

Vt (θ) = V0(θ)+

∫ t

0θ0(u)dB0

u +

∫ t

0θ1(u)dSu +

∫ t

0θ2(u)dPu, (3.4)


holds for any t a.s., the portfolio θ(t) is called self-financing.A self-financing portfolio θ(t) is called an arbitrage opportunity, provided that the three

properties V0(θ) = 0, VT (θ) ≥ 0 a.s. and P(VT (θ) > 0) > 0 hold. If there is no arbitrageopportunity, the market is called non-arbitrage. A self-financing portfolio is called riskless if thevalue process Vt (θ) does not include the martingale part.

We define normalized (or discounted) processes St , Pt and Vt (θ) by St = e−r t St , Pt = e−r t Ptand Vt (θ) = e−r t Vt (θ). It is not difficult to see that if the market is non-arbitrage, any risklessnormalized value process Vt (θ) is equal to V0(θ) for any 0 ≤ t ≤ T .

The next lemma is easily verified.

Lemma 3.1. Let (θ1(t), θ2(t)) be a pair of predictable processes such that∫ T

0 (|θ1|2+|θ2|

2)dt <∞ and V0 be an F0-measurable random variable. Set

θ0(t) = V0 +

∫ t

0θ1dSu +

∫ t

0θ2dPu − θ1(t)St − θ2(t)Pt . (3.5)

Then the triple θ(t) = (θ0(t), θ1(t), θ2(t)) is a self-financing portfolio. Further, its normalizedvalue process Vt (θ) satisfies

Vt (θ) = V0 +

∫ t

0θ1d Su +

∫ t

0θ2dPu . (3.6)

We will consider the problem of finding the pricing function F(t, x), which both the seller andthe buyer of the option could agree with. Discussions for the case where St is a diffusion and thecase where St is a jump–diffusion are quite different. In Sections 3.2 and 3.3, we shall considerthe case where St is a diffusion. The case where St has jumps will be discussed in Sections 3.4and 3.5.

3.2. The Black–Scholes partial differential equation

In this and the next subsections, we assume that the price process is a diffusion. Eq. (3.1) iswritten simply as

dSt = St b(t, St )dt + Stσ(t, St )dWt . (3.7)

We assume further that the market (B0t , St , Pt ) is non-arbitrage in this subsection.

Let θ(t) be a self-financing portfolio given in Lemma 3.1. We shall compute the normalizedvalue process Vt (θ) written by (3.6). Since St satisfies the above equation, St = e−r t St satisfies

dSt = e−r t St (b(t)− r)dt + e−r t Stσ(t)dWt , (3.8)

where b(t) = b(t, St ), σ (t) = σ(t, St ), etc. Set F(t, x) = e−r t F(t, x) and apply Ito’s formula.Then Pt := F(t, St ) satisfies

dPt = e−r t(

Ft + St b(t)Fx +12

S2t σ(t)

2 Fxx − r F)

dt + e−r t Stσ(t)Fx dWt . (3.9)

Then, dVt (θ) can be written as

dVt (θ) = θ1(t)dSt + θ2(t)dPt = β(t)dt + α(t)dWt ,


where

α(t) = e−r t (θ1(t)Stσ(t)+ θ2(t)Stσ(t)Fx ),

β(t) = e−r t(θ1(t)(St b(t)− Str)+ θ2(t)(Ft + St b(t)Fx +

12

S2t σ(t)

2 Fxx − r F)

).

Now, in order to eliminate the risk from the normalized value process Vt (θ), choose portfoliosθ1, θ2 such that α(t) = 0 holds for any t . For example, choose θ ′1(t) = Fx (t, St ), θ

′

2(t) = −1and define θ ′0(t) by (3.5) with setting V0 = 0. Set θ ′(t) = (θ ′0(t), θ

′

1(t), θ′

2(t)). Then the randomcomponent due to dWt is eliminated in dVt (θ

′). Further the drift coefficient β(t) is written as

β(t) = −e−r t(

Ft (t)+12

S2t σ(t)

2 Fxx (t)+ Str Fx (t)− r F(t)).

Then the riskless normalized value process Vt (θ′) should be equal to the initial value V0(θ

′),since the market is non-arbitrage. This implies that β(t) = 0 for all t . Since the support of thelaw of St is the whole space (0,∞), we have the following assertion.

Theorem 3.2 (Black–Scholes [1]). Suppose that the market (B0t , St , F(t, St )) is non-arbitrage.

Then the pricing function F(t, x), 0 ≤ t ≤ T, x > 0 satisfies the following backward partialdifferential equation.

Ft (t, x)+12

x2σ(t, x)2 Fxx (t, x)+ xr Fx (t, x)− r F(t, x) = 0, (3.10)

F(T, x) = h(x). (terminal condition)

We shall find a hedging portfolio for the option h(ST ). The triple θ ′(t) = (θ ′0(t), θ′

1(t), θ′

2(t))chosen above is a self-financing and riskless portfolio such that V0(θ

′) = 0. Then because ofnon-arbitrage, we have

θ ′0(t)B0t + Fx (t, St )St − F(t, St ) = Vt (θ

′) = 0.

This yields

θ ′0(t) = e−r t (F(t, St )− Fx (t, St )St ).

Since θ ′(t) is self-financing and Vt (θ′) = 0, we get from formula (3.4)∫ T

te−ru(F(u, Su)− Fx (u, Su)Su)dB0

u +

∫ T

tFx (u, Su)dSu −

∫ T

tdPu = 0.

Since∫ T

t dPu = h(ST )− F(t, St ), h(ST ) is represented by

h(ST ) = F(t, St )+

∫ T

te−ru(F(u, Su)− Fx (u, Su)Su)dB0

u +

∫ T

tFx (u, Su)dSu .

The above formula indicates that if we (buyer and seller of the option) hold the self-financingportfolio (θ ′0(t), θ

′

1(t)) for (B0t , St ), its value at the terminal time T coincides with the option

h(ST ). Consequently the option h(ST ) is completely hedged by the above portfolio. In otherwords, the portfolio (θ ′0(t), θ

′

1(t)) is a replica of the option. Therefore we can agree about F(t, St )

as the price of the option at t without any risk. For further informations on hedging problems, werefer to [20,27].


3.3. Equivalent martingale measure for diffusions

Let Q be another probability defined on the same measurable space (Ω ,F). If P and Q areequivalent (mutually absolutely continuous) on FT , the price process St can be regarded as astochastic process with respect to Q. An equivalent measure Q is called a martingale measure ifSt := e−r t St is a local martingale with respect to Q.

We shall construct a martingale measure. Set φ(t) = b(t,St )−rσ(t,St )

and define

αt = αt (φ) = exp∫ t

0φdWs −

12

∫ t

0|φ|2ds

.

Since φ(t) is a bounded process, αt is a positive martingale. Let Q = αT · P . By Girsanov’stheorem, Wφ

t := Wt −∫ t

0 φds is a Q-Brownian motion. Note that St satisfies (3.8). It is rewrittenas

dSt = Stσ(t)dWφt .

Therefore St is a martingale with respect to Q. Consequently Q is a martingale measure.Apply Theorem 2.10 to the case of a diffusion, i.e., ν ≡ 0. Then we find that with respect to

Q, St is a diffusion process with generator

L Q(t)F(x) =12

x2σ(t, x)2 F ′′(x)+ xr F ′(x).

The Black–Scholes backward partial differential equation (3.10) is written as(∂

∂t+ L Q(t)− r

)F = 0, (3.11)

F(T, x) = h(x).

It is known that for any continuous function h with polynomial growth, the solution of the aboveequation exists uniquely. It is given by

F(t, x) = E Q[e−r(T−t)h(S(t,x)T )], (3.12)

where S(s,x)t is the solution of Eq. (3.7) starting from x at time s. In other words,the Black–Scholes equation (3.11) is equal to Kolmogorov’s backward equation stated inTheorem 2.9 for the discounted transition function

P Qs,t f (x) = e−r(t−s)E Q

[ f (S(s,x)t )].

We will check this fact again in Section 3.5.The next theorem shows that the Black–Scholes equation for the pricing function F(t, x) is

a necessary and sufficient condition for the market (B0t , St , F(t, St )) to be non-arbitrage. Thus

the Black–Scholes equation is characterized as the unique non-arbitrage price of the Europeanoption with pay-off function h(x).

Theorem 3.3. Let F(t, x), 0 ≤ t ≤ T, x ∈ R+ be a C1,2-function with h(T, x) = h(x) andPt = F(t, St ). We set St = e−r t St and Pt = e−r t Pt as before. The following three statementsare equivalent.

(1) Pt is a Q-martingale.


(2) F satisfies the Black–Scholes equation.(3) The triple (B0

t , St , Pt ) is non-arbitrage.

Proof. We first show the equivalence of (1) and (2). We can rewrite Eq. (3.9) as

dPt = e−r t(∂

∂t+ L Q(t)− r

)F(t, St )dt + e−r t Stσ(t, St )dWφ

t . (3.13)

Hence if Pt is a martingale with respect to Q, then ( ∂∂t + L Q(t) − r)F(t, St ) = 0, proving the

Black–Scholes equation. Conversely if (2) is satisfied, then ( ∂∂t +L Q(t)− r)F = 0, so that Pt is

a martingale with respect to Q by formula (3.13).Next we will show that (1) implies (3). If (1) is satisfied, Vt (θ) is a continuous martingale.

If it is riskless, it should be a continuous process of finite variation. Then it should be equal toV0(θ), proving (3).

Finally suppose (3). We saw in Theorem 3.2 that F satisfies the Black–Scholes equation. ThenPt is a Q-martingale, in view of (3.13).

3.4. Merton’s integro-differential equation

In this subsection we assume that the price process St is a jump–diffusion determined by (3.1)and (3.2). We will again consider the triple (B0

t , St , Pt ) where Pt = F(t, St ) and F is a pricingfunction of the C1,2-class. We assume that the function F(t, x) is of linear growth with respectto x .

Let St = e−r t St and Pt = e−r t Pt as before. We have

dSt = e−r t St−(b(t, St−)− r)dt

+ e−r t St−

[σ(t, St−)dWt +

∫R0

g(t, St−, z)N (dtdz)

]. (3.14)

Apply Ito’s formula (Theorem 2.5) to the function F(t, x) = e−r t F(t, x). Then we have

dPt = e−r t( ∂∂t+ L(t)− r

)F(t, St−)dt

+ e−r t[

St−σ(t, St−)Fx (t, St−)dWt +

∫R0

GdN

], (3.15)

where L(t) is the generator of the jump–diffusion St , given by

L(t) f (x) =12

x2σ(t, x)2 fxx (x)+ xb(t, x) fx (x)

+

∫R0

f (x(1+ g(t, x, z)))− f (x)− xg(t, x, z) fx (x)ν(dz), (3.16)

and G = F(t, x(1 + g(t, x, z))) − F(t, x). Let θ(t) = (θ0(t), θ1(t), θ2(t)) be a self-financingportfolio for (B0

t , St , Pt ). Then the normalized value process Vt (θ) of (3.6) satisfies

dVt (θ) = β′(t)dt + α′(t)dWt +

∫R0

γ ′(t, z)N (dtdz),


where

α′(t) = e−r t [θ1(t)St−σ(t−)+ θ2(t)St−σ(t−)Fx],

β ′(t) = e−r t[θ1(t)(St−b(t−)− St−r)+ θ2(t)

(∂

∂t+ L(t)− r

)F

].

Here b(t−) = b(t, St−), σ (t−) = σ(t, St−).Now, in order to eliminate the continuous risk from the normalized value process Vt (θ), take

the portfolio θ ′′1 (t) = Fx (t, St−) and θ ′′2 (t) = −1. Then we have α′(t) = 0. Further, we canrewrite β ′(t) explicitly. In fact, the normalized value process is written as

Vt (θ′′) = V0 +

∫ t

0e−rs A(s)F(s, Ss−)ds +

∫ t

0

∫R0

γ ′(r, z)N (drdz),

where

A(t)F(t, x) = Ft (t, x)+12

x2σ(t, x)2 Fxx (t, x)+ xr Fx (t, x)− r F(t, x)

+

∫R0

(F(t, x + xg(t, x, z))− F(t, x)− xg(t, x, z)Fx (t, x)

)dν(z). (3.17)

We have thus eliminated the continuous risk concerned with dWt . However, we cannoteliminate γ ′(t, z), i.e., the jump risk by any portfolio (θ1(t), θ2(t)). Then, the jump risk shouldbe neutral to the seller and the buyer of the option. Hence the normalized value process Vt (θ

′′)

should be risk neutral, i.e., it should be a local martingale. Then we have A(t)F(t, St−) = 0 forany t . Merton [29] adopted such F(t, x) as the pricing function of the option.

Theorem 3.4 (Merton [29]). Let (B0t , St , F(t, St )) be the triple where St is a jump–diffusion

determined by SDE (3.1). Then there exists a self-financing portfolio which eliminates thecontinuous risk and makes the jump risk neutral, if and only if the pricing function F(t, x)satisfies the backward integro-differential equation

A(t)F(t, x) = 0, ∀(t, x) ∈ [0, T )× (0,∞), (3.18)

F(T, x) = h(x), (terminal condition).

We will see in Proposition 3.9 that the above pricing function F makes (B0, St , F(t, St ))

non-arbitrage.A self-financing portfolio (θ ′′0 (t), θ

′′

1 (t), θ′′

2 (t)) which hedges the continuous risk is given by

θ ′′0 (t) = e−r t (F(t, St )− Fx (t, St−)St ), θ ′′1 (t) = Fx (t, St−), θ ′′2 (t) = −1.

It takes the same form as the portfolio in the diffusion case.

3.5. Equivalent martingale measures for jump–diffusions

In the previous subsection, we reviewed the approach by Merton for pricing options. Weobtained Merton’s integro-differential equation (3.18) for the pricing function F(t, x). In thissubsection, we will show the existence and uniqueness of the solution of Merton’s equation byusing an equivalent martingale measure.

We will show that there are infinitely many equivalent martingale measures for ajump–diffusion, in contrast to the unique martingale measure for a diffusion. Let Q be an


equivalent probability measure and let αt be the Radon–Nikodym density dQdP |Ft . We saw in

Section 2.3 that it is represented as the unique solution of a SDE dαt = αt−dZ t with the initialcondition α0 = 1, where Z t is a local martingale written as

Z t =

∫ t

0φ(s)dWs +

∫ t

0

∫ψ(s, z)N (dsdz),

with φ ∈ Φ2loc, ψ ∈ Ψloc. We denote it as αt (φ, ψ).

Lemma 3.5. ([24]) St is a local martingale with respect to Q if and only if ψ(t, z)g(t, St−, z) ∈Ψ1

loc and satisfies a.e. dt ⊗ P,

b(t, St−)+ φ(t)σ (t, St−)+

∫R0

ψ(t, z)g(t, St−, z)ν(dz) = r. (3.19)

Proof. The measure Q is a martingale measure if and only if αt St is a local martingale withrespect to P . It holds

d(αt St ) = St dαt + αt−dSt + d[α, S]t = αt− St−(dZ t + dYt + d[Z , Y ]t ).

Since [Z , Y ]t =∫ t

0 φσds +∫ t

0

∫ψgdN , we have

dZ t + dYt + d[Z , Y ]t = dZ t + σ(t−)dWt + (b(t−)− r)dt + φ(t)σ (t−)dt +∫

R0

ψgdN .

The above is a local martingale if and only if ψg ∈ Ψ1loc and the equality

(b(t−)− r)+ φ(t)σ (t−)+∫

R0

ψ(t, z)g(t, St−, z)dν = 0, dt a.e.

holds.

If ν = 0, Eq. (3.19) is equivalent to b(t−) + φ(t)σ (t−) = r, dt a.e.. Then the solutionφ(t) exists uniquely as an element of Φ2

loc. It is a bounded function dt a.e. Hence an equivalentmartingale measure exists uniquely. On the other hand, if ν 6= 0, there exist infinitely many pairs(φ(t), ψ(t)) such that φ,ψ and

∫ψ(t, z)2ν(dz) are bounded and satisfy Eq. (3.19). Therefore

we have

Proposition 3.6. (1) If ν = 0, a martingale measure exists uniquely.(2) If ν 6= 0, there are infinitely many martingale measures.

Now let ψ(t, x, z) be a function such that∫ψ(t, x, z)2ν(dz) is bounded with respect to t, x .

There exists a bounded φ(t, x) satisfying

b(t, x)+ φ(t, x)σ (t, x)+∫

R0

ψ(t, x, z)g(t, x, z)ν(dz) = r. (3.20)

Then the local martingale αt = αt (φ, ψ) is a martingale and Q = αT ·P is a martingale measure.We denote by Q the set of all martingale measures Q = αT (φ, ψ) · P such that the pair (φ, ψ)satisfies the above property.

The next proposition follows from Theorem 2.10 since bφ,ψ (t, x) = r holds by (3.20).


Proposition 3.7. With respect to Q ∈ Q, St is a jump–diffusion process. Its generator is givenby

L Q(t) f (x) =12

x2σ(t, x)2 f ′′(x)+ xr f ′(x)

+

∫R0

f (x + xg(t, x, z))− f (x)− xg(t, x, z) f ′(x)

(1+ ψ(t, x, z))ν(dz). (3.21)

A simple martingale measure is obtained in the case ψ ≡ 0. Then φ0(t) = (b(t)− r)/σ (t) isthe unique solution of Eq. (3.20). We denote the corresponding martingale measure by Q0. Withrespect to Q0, St is a jump–diffusion. Its generator L Q0(t) is given by (3.21) where ψ(t, x, z) =0. Therefore ( ∂

∂t +L Q0(t)− r)F coincides with A(t)F given by (3.17). Consequently Merton’sintegro-differential equation (3.18) is rewritten as(

∂

∂t+ L Q0(t)− r

)F = 0, (3.22)

F(T, x) = h(x).

Theorem 3.8. Let S(s,x)t be the solution of Eq. (3.1) starting from x at time s. If it is a geometricLevy process, then

F(t, x) := E Q0 [e−r(T−t)h(S(t,x)T )] (3.23)

is a C1,2-function. It is the unique C1,2-solution of the integro-differential equation (3.22).

Proof. The solution is represented as S(t,x)u = x exp(Zu− Z t ), where Z t is a homogeneous Levyprocess. Since the law of the Levy process ZT − Z t − r(T − t) has a smooth density pt,T (z),the law of er(T−t)S(t,x)T has also a density given by pt,T (log z − log x)z−1. It is a C1,2 functionof (t, x). The function F of (3.23) is then written as F(t, x) =

∫h(z)pt,T (log z − log x)z−1dz.

Consequently it is a C1,2-function.We will show that the function F satisfies Eq. (3.22). Let St , t ≥ 0 be any solution starting at

time 0. Observe that Pt = e−r t F(t, St ) is a Q0-martingale because we have for any t > s,

E Q0 [e−r t F(t, St )|Fs] = E Q0 [e−rT h(ST )|Fs],

in view of the Markov property of St . Apply Ito’s formula to e−r t F(t, St ) under the measure Q0.Then we have, similarly as (3.15)

dPt = e−r t( ∂∂t+ L Q0(t)− r

)F(t, St )dt + dM0

t ,

where M0t is a local Q0-martingale. Since Pt is also a Q0-martingale, the drift part of the above

is equal to 0 a.s. Then we get ( ∂∂t + L Q0(t)− r)F = 0.

We show the uniqueness of the solution. Let F ′(t, x) be any C1,2-solution of Eq. (3.22)with linear growth. Set P ′t = e−r t F ′(t, St ). Applying Ito’s formula and noting that F ′ is asolution of Eq. (3.22), we see that P ′t is a local martingale. It holds P ′T = h(ST ). Then wehave P ′T = PT . Since P ′t and Pt are local martingales, we have Pt = P ′t for any t . Thereforewe get F(t, St ) = F ′(t, St ) a.s. for any t . Since the support of the law of St is the whole spaceR+ = (0,∞), we get F(t, x) = F(t, x) for all x ∈ R+. The proof is complete.


Remark. In the case where the coefficients b, σ, g depend on x , we cannot apply the abovetheorem directly. However, if these coefficients are smooth functions, say C∞, the solution S(t,x)uis a C∞-function of x , in view of the theory of stochastic flows [23]. Thus if the value function hin (3.23) is a C2-function, F(t, x) is also a C1,2-function and the assertion of the lemma is valid.

If the value function h is not smooth as for call or put options, we may need Malliavin calculuson the Wiener–Poisson space [6]. Discussions are close to Watanabe on the Wiener space [41].Here we give a very rough argument. The solution X = S(t,x)u is a smooth random variable(∈ D∞) for any t < u, x . Let Φ be a tempered distribution. We may define the compositionΦ S(t,x)u as a generalized random variable in D′∞ (dual space of D∞) with parameter s, t, x .Further it can be shown that 〈Φ S(t,x)u ,G〉 is infinitely differentiable with respect to x , ifG ∈ D∞. In particular 〈ΦS(t,x)u , 1〉 is smooth with respect to x . It coincides with the expectationE[Φ S(t,x)u ] if Φ is a function of polynomial growth. Details will be discussed elsewhere [25].

Finally we shall consider other martingale measures Q ∈ Q. Let F(t, x), 0 ≤ t ≤ T, x ∈ R+

be a C1,2-function such that F(T, x) = h(x) and let Pt = F(t, St ). Apply Ito’s formula again toe−r t F(t, St ). With respect to the measure Q, Pt = e−r t Pt satisfies similarly as in (3.15),

dPt = e−r t( ∂∂t+ L Q(t)− r

)F(t, St )dt + dMt ,

where L Q is the operator defined by (3.21) and Mt is a local Q-martingale. Then Pt is a localQ-martingale if and only if ( ∂

∂t + L Q(t)− r)F = 0.The following is verified similarly as Theorem 3.3.

Theorem 3.9. Let Q be any martingale measure in Q. Then the stochastic process Pt =

e−r t F(t, St ) is a Q-martingale if and only if F satisfies(∂

∂t+ L Q(t)− r

)F = 0. (3.24)

Further, if F satisfies (3.24), then the triple (B0t , St , F(t, St )) is non-arbitrage.

The above theorem tells us that in case of a jump–diffusion, there are infinitely many pricingfunctions F for which the triple (B0

t , St , F(t, St )) is non-arbitrage. For any martingale measureQ ∈ Q, a solution of the equation ( ∂

∂t + L Q(t) − r)F = 0 with the terminal conditionF(T, x) = h(x) gives us a non-arbitrage price of the European option with pay-off functionh. On the other hand, in case of a diffusion, such F is unique and it is the solution of theBlack–Scholes equation, in view of Theorem 3.3.

Concerning the choice of one out of many martingale measures Q, there are some alternativeproposals: super-hedging, utility maximization, minimal variance hedging, use of a minimalentropy martingale measure and so on. For these subjects, we refer to R. Cont-P. Tankov [2]and the references therein.

Acknowledgement

Marc Yor read carefully preliminary versions of this paper and gave me many usefulcomments and advices. I should like to express my gratitude to him.

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